L.Brugnano and J.R.Cash For a number of years this special issue of JNAIAM has been devoted to the proceedings of the ICNAAM Conference series. In particular the seventh conference, held in Rethymno, Crete (GR), from 18th to 22th September 2009, celebrates the 60th birthday of Professor Ernst Hairer. As is well known, Ernst is one of the leading experts in the numerical solution of ODEs. He has contributed substantially to the field, both in the theoretical analysis of numerical methods, and from the point of view of software development. He is coauthor of a number of monographs on this topic, as well as of some of the most reliable codes for stiff ODEs, based on Radau IIA formulae. One of the fields where he has been very involved in the last few years is that of geometric numerical integration, where he is coauthor, with Christian Lubich and Gerard Wanner, of one of the most comprehensive monographs on the subject.
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Thursday, January 27. 2011
G. Kitzhofer, O. Koch, G. Pulverer, Ch. Simon, and E.B. Weinm¨uller2 Institute for Analysis and Scientific Computing (E101), Vienna University of Technology, Wiedner Hauptstrasse 8–10, A1040 Wien, Austria Received 21 January, 2010; accepted in revised form 23 March, 2010 Abstract: Our aim is to provide the open domain MATLAB code bvpsuite for the efficient numerical solution of boundary value problems in ordinary differential equations. Motivated by applications, we are especially interested in designing a code whose scope is appropriately wide, including fully implicit problems of mixed orders, parameter dependent problems, problems with unknown parameters, problems posed on semiinfinite intervals, eigenvalue problems and differential algebraic equations of index 1. Our main focus is on singular boundary value problems in which singularities in the differential operator arise. We first briefly recapitulate the analytical properties of singular systems and the convergence behavior of polynomial collocation used as a basic solver in the code for both singular and regular ordinary differential equations and differential algebraic equations. We also discuss the a posteriori error estimate and the grid adaptation strategy implemented in our code. Finally, we describe the code structure and present the performance of the code which has been equipped with a graphical user interface for an easy use.
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Thursday, January 27. 2011
Robert D. Skeel2 Department of Computer Science, Purdue University, West Lafayette, Indiana 479072107, U.S.A. Received 26 February, 2010; accepted in revised form 11 May, 2010 Abstract: Markov chain Monte Carlo methods are very popular for computing expectations. Their efficiency and reliability are subject to two significant drawbacks. The first is the correlation between successive samples. This reduces efficiency and frustrates variance estimation. The second drawback is the dependence on starting values, which leads to discarding a large initial set of “atypical” samples. The process of running the Monte Carlo method until getting an adequate starting value is called equilibration. Associated with this are two practical problems. One is to detect the onset of equilibration so that production may begin. The other is to characterize what it means to be equilibrated so that there might be a better understanding of how to initialize the equilibration process to reduce its running time. This article examines the statistical error of Monte Carlo method and proposes a definition of what it means to be equilibrated, which corresponds exactly to what is needed in practice and which is amenable to mathematical analysis.
c 2010 European Society of Computational Methods in Sciences and Engineering Keywords: markov chain monte carlo methods, equilibration, correlation, markov process, molecular simulation, sampling, mixing, finite state, statistical inefficiency Mathematics Subject Classification: 65C40, 60J22, 82B80
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Thursday, January 27. 2011
Fathalla A. Rihan2 Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, 17551, UAE Received 11 October, 2009; accepted in revised form 21 January, 2010 Abstract: In this short paper, we investigate sensitivity and robustness of neutral delay differential models to small perturbations in the parameters that occur in the models, using variational approach. The technique provides a guidance for the modelers to determine the most informative data for a specific parameter. It may also help modelers to select the best fit model to the observations.
c 2010 European Society of Computational Methods in Sciences and Engineering Keywords: Adjoint; Neutral delay differential equations; Sensitivity; Timelag Mathematics Subject Classification: 93A13; 93A30; 34K05; 34K28; 34K40; 45G15; 47N60; 65Y20; 65L2.
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Thursday, January 27. 2011
Abstract: The characterizations of Bseries of symplectic and energy preserving integrators are wellknown. The graded Lie algebra of Bseries of modified vector fields include the Hamiltonian and energy preserving cases as Lie subalgebras, these spaces are relatively well understood. However, two other important classes are the integrators which are conjugate to Hamiltonian and energy preserving methods respectively. The modified vector fields of such methods do not form linear subspaces and the notion of a grading must be reconsidered. We suggest to study these spaces as filtrations, and viewing each element of the filtraton as a vector bundle whose typical fiber replaces the graded homogeneous components. In particular, we shall study properties of these fibers, a particular result is that, in the energy preserving case, the fiber of degree n is a direct sum of the nth graded component of the Hamiltonian and energy preserving space. We also give formulas for the dimension of each fiber, thereby providing insight into the range of integrators which are conjugate to symplectic or energy preserving.
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Thursday, January 27. 2011
E. Hairer3 Universit´e de Gen`eve, Section de Math´ematiques, 24 rue du Li`evre, CH1211 Gen`eve 4, Switzerland Received 15 October, 2009; accepted in revised form 21 March, 2010 Abstract: We propose a modification of collocation methods extending the ‘averaged vector field method’ to high order. These new integrators exactly preserve energy for Hamiltonian systems, are of arbitrarily high order, and fall into the class of Bseries integrators. We discuss their symmetry and conjugatesymplecticity, and we compare them to energypreserving composition methods.
c 2010 European Society of Computational Methods in Sciences and Engineering Keywords: Hamiltonian systems, energypreserving integrators, Bseries, Runge–Kutta methods, collocation, conjugatesymplecticity, symmetry, Gaussian quadrature. Mathematics Subject Classification: 65P10, 65L06
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Thursday, January 27. 2011
C. W. Gear,2 D. Givon and I. G. Kevrekidis Department of Chemical Engineering, Princeton University, Princeton, NJ, USA Received 10 January, 2010; accepted in revised form 20 March, 2010 Abstract: The persistently fast evolutionary behavior of certain differential systems may have intrinsically slow features. We consider systems whose solution trajectories are slowly changing distributions and assume that we do not have access to the equations of the system, only to a simulator or legacy code that performs stepbystep time integration of the system. We characterize the set of all possible instantaneous solutions by points on a lowdimensional virtual slow manifold (VSM) and show how, when there is a sufficiently large gap between the time scales of the fast and slow behaviors, we can restrict the fast behavior observed numerically through simulation to the VSM and perform useful computations more efficiently there.
c 2010 European Society of Computational Methods in Sciences and Engineering Keywords: Projective Integration, Steady State, Legacy Codes, Equation Free Mathematics Subject Classification: 65L05,65M99
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Thursday, January 27. 2011
B. V. Faleichik2 Computational Mathematics Department, Faculty of Applied Mathematics and Computer Science, Belarusian State University, 220030 Minsk, Belarus Received October 21, 2009; accepted in revised form January 19, 2010. Abstract: Currently there are two general ways to solve stiff differential equations numerically. The first approach is based on implicit methods and the second uses explicit stabilized Runge–Kutta methods, also known as Chebyshev methods. Implicit methods are great for very stiff problems of not very large dimension, while stabilized explicit methods are efficient for very big systems of not very large stiffness and real spectrum. In this paper we describe methods which are explicit and are capable of solving stiff systems with complex eigenvalues of Jacobi matrix.
c 2010 European Society of Computational Methods in Sciences and Engineering Keywords: Stiff problems, explicit methods, collocation methods, iterated RungeKutta methods, linear analysis of convergence. Mathematics Subject Classification: 65L05, 65L06, 65L20
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Thursday, January 27. 2011
J. C. Butcher2 Department of Mathematics, University of Auckland, Auckland, New Zealand Received February 8, 2010; accepted in revised form February 22, 2010. Abstract: Bseries, together with the algebraic system which underpins them, are essential tools in the study of properties of numerical methods for evolutionary problems. This paper surveys the properties of these constructs and relates the theory to applications in numerical analysis.
c 2010 European Society of Computational Methods in Sciences and Engineering Keywords: Bseries, Runge–Kutta methods, order conditions, effective order, singlyimplicit methods. Mathematics Subject Classification: 65L05, 65L06
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Thursday, January 27. 2011
Luigi Brugnano3 Dipartimento di Matematica “U.Dini”, Universit`a di Firenze Viale Morgagni 67/A, I50134 Firenze, Italy Felice Iavernaro4 Dipartimento di Matematica, Universit`a di Bari Via Orabona 4, I70125 Bari, Italy Donato Trigiante5 Dipartimento di Energetica “S.Stecco”, Universit`a di Firenze Via Lombroso 6/17, I50134 Firenze, Italy Received October 25, 2009; accepted in revised form April 15, 2010. Abstract: Recently, a new family of integrators (Hamiltonian Boundary Value Methods) has been introduced, which is able to precisely conserve the energy function of polynomial Hamiltonian systems and to provide a practical conservation of the energy in the nonpolynomial case. We settle the definition and the theory of such methods in a more general framework. Our aim is on the one hand to give account of their good behavior when applied to general Hamiltonian systems and, on the other hand, to find out what are the optimal formulae, in relation to the choice of the polynomial basis and of the distribution of the nodes. Such analysis is based upon the notion of extended collocation conditions and the definition of discrete line integral, and is carried out by looking at the limit of such family of methods as the number of the so called silent stages tends to infinity. c
2010 European Society of Computational Methods in Sciences and Engineering Keywords: Hamiltonian problems, exact conservation of the Hamiltonian, energy conservation, Hamiltonian Boundary Value Methods, HBVMs, discrete line integral.
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Thursday, January 27. 2011
Pierluigi Amodio2, Giuseppina Settanni3 Dipartimento di Matematica, Universit`a di Bari, I70125 Bari, Italy Received 22 February, 2010; accepted in revised form 13 April, 2010 Abstract: The numerical solution of second order ordinary differential equations with initial conditions is here approached by approximating each derivative by means of a set of finite difference schemes of high order. The stability properties of the obtained methods are discussed. Some numerical tests, reported to emphasize pros and cons of the approach, motivate possible choices on the use of these formulae.
c 2010 European Society of Computational Methods in Sciences and Engineering Keywords: Second order Initial Value Problems, finite difference schemes, Boundary Value Methods, absolute stability.
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